MODULE 04Policy & welfare loss

Taxes & Deadweight Loss

Baseline · $10 tax · 43.75 unitsAll modules

Module 04 · Theory

Taxes, Price Controls, and Deadweight Loss

Mankiw · Ch. 6 & 8 · Supply, demand, and government policies; the costs of taxation

A tax is a wedge between what buyers pay and what sellers keep. A price control is a rule about one of those prices. Both change the quantity traded, and both create a gap between the value of the marginal unit and its cost. That gap is deadweight loss.

The tax wedge

With a tax of T per unit, buyers pay P_B and sellers keep P_S, withP_B − P_S = T. The tax raises the buyer's price and lowers the seller's price, so quantity falls. The government collectsT × Q_T, shown as the revenue rectangle inside the wedge.

Incidence depends on elasticity

The legal assignment of the tax does not determine who bears it. The economic burden follows the slopes: the steeper, less responsive side bears the larger share. In this linear model, buyers bear b / (b + d) and sellers beard / (b + d). Make demand steeper and the buyer share rises.

Deadweight loss

The triangle between demand and supply from Q_T toQ* is the surplus lost because those units no longer trade. It grows with the square of the tax. Revenue first rises with the tax, but DWL rises faster and never comes back down.

The Laffer curve

Revenue is T × Q_T. A higher rate collects more per unit but shrinks the tax base. In the linear model the two effects produce a parabola: revenue peaks at T = $40 and then falls. The peak is not a welfare recommendation; DWL keeps rising on both sides of it.

Price ceilings and floors

A ceiling below equilibrium makes sellers the short side: quantity supplied is all that trades, and a shortage appears. A floor above equilibrium makes buyers the short side: quantity demanded is all that trades, and a surplus appears. A control on the non-binding side of equilibrium changes nothing.

Formulas

Q_T = (100 − 20 − T) / 1.6

P_B = 100 − Q_T

P_S = 20 + 0.6Q_T

Revenue = T × Q_T

DWL = ½ T (Q* − Q_T)

Baseline: Q* = 50, P* = $50. At T = $10: Q_T = 43.75, P_B = $56.25, P_S = $46.25, revenue = $437.50, DWL = $31.25.

Try this

01Set the $10 textbook tax. Buyers bear 62.5% of it and sellers bear 37.5%, even though the tax is legally imposed on no one in particular.
02Raise demand steepness to 2.0×. Buyers become less responsive and their share of the tax rises.
03Jump to the Laffer peak at $40. Revenue reaches $1,000, but DWL is already $500.
04Switch to a $40 ceiling or a $60 floor. The short side of the market sets the quantity, and the lost trades become DWL.

Live policyTax $10 · Q 43.8

Tax wedge

Consumer surplusProducer surplusTax revenueDeadweight lossEquilibrium E

Drag the white handle at the top of the tax wedge, or use the sliders, to change the tax. Focus the handle and use the arrow keys (Shift for larger steps).

Live readout

The wedge, split and lost

Tax wedge · On the rising side

Tax revenue

$438

T × Q_t

Surplus transferred to the government.

Deadweight loss

$31

surplus destroyed

The triangle between demand and supply.

Consumer surplus

$957

after the tax

Area below demand and above the buyer's price.

Producer surplus

$574

after the tax

Area above supply and below the seller's price.

Tax incidence

economic, not statutory

Buyers 62.5%

+$6.25 per unit

Sellers 37.5%

−$3.75 per unit

The steeper, less responsive side of the market bears the larger share. The legal assignment of the tax does not change this split.

Buyer price / seller price

The gap between them is the unit tax.

$56.25 / $46.25

Welfare identity

CS + PS + revenue + DWL stays at the efficient total.

$2,000

Quantity traded

6.3 units no longer trade.

43.8units

Laffer reference

Peak revenue is $1,000 at a $40 tax.

$438

Laffer tracker

Revenue versus the tax rate

Peak $40 · $1,000

Revenue rises while a higher rate outweighs the shrinking tax base, peaks when the two effects balance, then falls. Deadweight loss keeps rising past the peak.